Question

Growth and population The Treasury is currently(estimating the growth level for Australia.)They are concerned about population growth and they have ask you to help them to prepare a report that accounts for an increase in the population size over time. You remember your time at UNSW and decide to use the to prepare your report.

The ABS forecast that population will growth at a constant rate n, where n>0.
When this assumption is made, the capital accumulation equation (in per capita terms) assumes the following format: Δkₜ=syₜ−(n+δ)kₜ,
where s is the savings rate and δ is the capital depreciation rate. In your report you need to answer the following questions: 1. Find the steady-state level of income per capita assuming the production function follows the standard CobbDouglas production function. Show your work, step-by-step.

Answer 1

The steady-state level of income per capita in a Cobb-Douglas production function is found by setting the change in capital per capita to zero and solving the resultant equation.

Step-by-step explanation:

Finding the Steady-State Level of Income Per Capita

To find the steady-state level of income per capita with a Cobb-Douglas production function, we assume the production function has the form yₜ = Aₜkₜ^α, where yₜ is the income per capita, kₜ is the capital per capita, Aₜ is the level of technology, and α is the output elasticity of capital, which is between 0 and 1. In the steady state, the change in capital per capita Δkₜ is zero. From the given capital accumulation equation Δkₜ = syₜ−(n+δ)kₜ, setting Δkₜ equal to zero gives:

1. 0 = sAₜkₜ^α - (n + δ)kₜ
2. solve for kₜ by dividing both sides by s and factoring out kₜ:
3. kₜ(sAₜkₜ^(α-1) - (n + δ)) = 0
4. Since kₜ cannot be zero in this context (no capital would imply no output), we must have sAₜkₜ^(α-1) = n + δ.
5. Rearrange to solve for kₜ:
6. kₜ = (sAₜ / (n + δ))^(1 / (1−α))
7. This represents the steady-state capital per capita.
8. To find the steady-state income per capita, substitute the steady-state kₜ back into the production function:
9. yₜ = Aₜ((sAₜ / (n + δ))^(1 / (1−α)))^α
10. Simplify this expression to find the steady-state level of income per capita.

Understanding the steady-state income per capita is crucial as it represents a level where the economy can sustain itself without further accumulation or depletion of resources. It's an indicator of sustained economic growth and the well-being of a nation's population.